The Friedmann equations are differential equations: insofar they are a correct description of what happens expansion and contraction have to be continuous. Were $a(t)$ to discontinuously change value they would become ill-posed and not predict what the universe does next.
In fact, deep down General Relativity is based on the idea that space-time is a metric manifold $(M,g)$ where the metric is $g$ differentiable and the manifold $M$ locally always can be mapped to Minkowski space. Discontinuous jumps of $a(t)$ implies nondifferentiability of $g$, and were it to correspond to more than coordinate shenanigans then it would break the manifold topological structure.
Over what range this model is applicable remains to be seen, but at least large-scale it seems to be doing fine.